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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>One solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\vec{y}^{(1)}(t)=\vec{\xi}^{(1)} e^{r_1 t}=({\bf a}+i{\bf b}) e^{(\lambda+i\mu) t}=e^{\lambda t} ({\bf a} \cos \mu t-{\bf b}\sin \mu t)+i e^{\lambda t}({\bf b} \cos \mu t+{\bf a}\sin \mu t),
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(e^{(\lambda+i\mu) t}=e^{\lambda t} (\cos \mu t+ i \sin \mu t )\text{.}\)</span> The other solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\vec{y}^{(2)}(t)=\vec{\xi}^{(2)} e^{r_2 t}=({\bf a}-i{\bf b}) e^{(\lambda-i\mu) t}=e^{\lambda t} ({\bf a} \cos \mu t-{\bf b}\sin \mu t)-i e^{\lambda t}({\bf b} \cos \mu t+{\bf a}\sin \mu t).
\end{equation*}
</div>
<p class="continuation">Linear combination of <span class="process-math">\(\vec{y}^{(1)}\)</span> and <span class="process-math">\(\vec{y}^{(2)}\)</span> is still a solution. Thus</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\bf x}^{(1)}=\frac{1}{2} (\vec{y}^{(1)}+\vec{y}^{(2)})=e^{\lambda t} ({\bf a} \cos \mu t-{\bf b}\sin \mu t),\quad {\bf x}^{(2)}=\frac{1}{2 i} (\vec{y}^{(1)}-\vec{y}^{(2)})=e^{\lambda t}({\bf b} \cos \mu t+{\bf a}\sin \mu t)
\end{equation*}
</div>
<p class="continuation">are two real solutions. Note, they are also linear independent.</p>
<span class="incontext"><a href="sec6_3.html#p-267" class="internal">in-context</a></span>
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